glamlassoS {glamlasso}

Penalization in Large Scale Generalized Linear Array Models

Description

Efficient design matrix free procedure for fitting a special case of a generalized linear model with array structured response and partially tensor structured covariates. See Lund and Hansen, 2019 for an application of this special purpose function.

Usage

glamlassoS(X, 
           Y,
           V, 
           Z = NULL,
           family = "gaussian",
           penalty = "lasso",
           intercept = FALSE,
           weights = NULL,
           betainit = NULL,
           alphainit = NULL,
           nlambda = 100,
           lambdaminratio = 1e-04,
           lambda = NULL,
           penaltyfactor = NULL,
           penaltyfactoralpha = NULL,
           reltolinner = 1e-07,
           reltolouter = 1e-04,
           maxiter = 15000,
           steps = 1,
           maxiterinner = 3000,
           maxiterouter = 25,
           btinnermax = 100,
           btoutermax = 100,
           iwls = "exact",
           nu = 1)

Arguments

X

A list containing the tensor components (2 or 3) of the tensor design matrix. These are matrices of sizes n_i \times p_i.

Y

The response values, an array of size n_1 \times\cdots\times n_d. For option family = "binomial" this array must contain the proportion of successes and the number of trials is then specified as weights (see below).

V

The weight values, an array of size n_1 \times\cdots\times n_d.

Z

The non tensor structrured part of the design matrix. A matrix of size n_1 \cdots n_d\times q. Is set to NULL as default.

family

A string specifying the model family (essentially the response distribution). Possible values are "gaussian", "binomial", "poisson", "gamma".

penalty

A string specifying the penalty. Possible values are "lasso", "scad".

intercept

Logical variable indicating if the model includes an intercept. When intercept = TRUE the first coulmn in the non-tensor design component Z is all 1s. Default is FALSE.

weights

Observation weights, an array of size n_1 \times \cdots \times n_d. For option family = "binomial" this array must contain the number of trials and must be provided.

betainit

The initial parameter values. Default is NULL in which case all parameters are initialized at zero.

alphainit

A q\times 1 vector containing the initial parameter values for the non-tensor parameter. Default is NULL in which case all parameters are initialized at 0.

nlambda

The number of lambda values.

lambdaminratio

The smallest value for lambda, given as a fraction of \lambda_{max}; the (data derived) smallest value for which all coefficients are zero.

lambda

The sequence of penalty parameters for the regularization path.

penaltyfactor

An array of size p_1 \times \cdots \times p_d. Is multiplied with each element in lambda to allow differential shrinkage on the coefficients.

penaltyfactoralpha

A q \times 1 vector multiplied with each element in lambda to allow differential shrinkage on the non-tensor coefficients.

reltolinner

The convergence tolerance for the inner loop

reltolouter

The convergence tolerance for the outer loop.

maxiter

The maximum number of inner iterations allowed for each lambda value, when summing over all outer iterations for said lambda.

steps

The number of steps used in the multi-step adaptive lasso algorithm for non-convex penalties. Automatically set to 1 when penalty = "lasso".

maxiterinner

The maximum number of inner iterations allowed for each outer iteration.

maxiterouter

The maximum number of outer iterations allowed for each lambda.

btinnermax

Maximum number of backtracking steps allowed in each inner iteration. Default is btinnermax = 100.

btoutermax

Maximum number of backtracking steps allowed in each outer iteration. Default is btoutermax = 100.

iwls

A string indicating whether to use the exact iwls weight matrix or use a kronecker structured approximation to it.

nu

A number between 0 and 1 that controls the step size \delta in the proximal algorithm (inner loop) by scaling the upper bound \hat{L}_h on the Lipschitz constant L_h (see Lund et al., 2017). For nu = 1 backtracking never occurs and the proximal step size is always \delta = 1 / \hat{L}_h. For nu = 0 backtracking always occurs and the proximal step size is initially \delta = 1. For 0 < nu < 1 the proximal step size is initially \delta = 1/(\nu\hat{L}_h) and backtracking is only employed if the objective function does not decrease. A nu close to 0 gives large step sizes and presumably more backtracking in the inner loop. The default is nu = 1 and the option is only used if iwls = "exact".

Details

Given the setting from glamlasso we consider a model where the tensor design component is only partially tensor structured as

X = [V_1X_2^\top\otimes X_1^\top,\ldots,V_{n_3}X_2^\top\otimes X_1^\top]^\top.

Here X_i is a n_i\times p_i matrix for i=1,2 and V_i is a n_1n_2\times n_1n_2 diagonal matrix for i=1,\ldots,n_3.

Letting Y denote the n_1\times n_2\times n_3 response array and V the n_1\times n_2\times n_3 weight array containing the diagonals of the V_is, the function glamlassoS solves the PMLE problem using Y, V, X_1, X_2 and the non-tensor component Z as input.

Value

An object with S3 Class "glamlasso".

spec

A string indicating the model family and the penalty.

beta

A p_1\cdots p_d \times nlambda matrix containing the estimates of the parameters for the tensor structured part of the model (beta) for each lambda-value.

alpha

A q \times nlambda matrix containing the estimates of the parameters for the non tensor structured part of the model (alpha) for each lambda-value. If intercept = TRUE the first row contains the intercept estimate for each lambda-value.

lambda

A vector containing the sequence of penalty values used in the estimation procedure.

df

The number of nonzero coefficients for each value of lambda.

dimcoef

A vector giving the dimension of the model coefficient array \beta.

dimobs

A vector giving the dimension of the observation (response) array Y.

Iter

A list with 4 items: bt_iter_inner is total number of backtracking steps performed in the inner loop, bt_enter_inner is the number of times the backtracking is initiated in the inner loop, bt_iter_outer is total number of backtracking steps performed in the outer loop, and iter_mat is a nlambda \times maxiterouter matrix containing the number of inner iterations for each lambda value and each outer iteration and iter is total number of iterations i.e. sum(Iter).

Author(s)

Adam Lund

Maintainer: Adam Lund, [e-mail]

References

Lund, A., M. Vincent, and N. R. Hansen (2017). Penalized estimation in large-scale generalized linear array models. Journal of Computational and Graphical Statistics, 26, 3, 709-724. url = https://doi.org/10.1080/10618600.2017.1279548.

Lund, A. and N. R. Hansen (2019). Sparse Network Estimation for Dynamical Spatio-temporal Array Models. Journal of Multivariate Analysis, 174. url = https://doi.org/10.1016/j.jmva.2019.104532.

Examples

##size of example
n1 <- 65; n2 <- 26; n3 <- 13; p1 <- 13; p2 <- 5; 

##marginal design matrices (tensor components)
X1 <- matrix(rnorm(n1 * p1), n1, p1)
X2 <- matrix(rnorm(n2 * p2), n2, p2)
X <- list(X1, X2)
V <- array(rnorm(n3 * n2 * n1), c(n1, n2, n3))

##gaussian example
Beta <- array(rnorm(p1 * p2) * rbinom(p1 * p2, 1, 0.1), c(p1 , p2))
Mu <- V * array(RH(X2, RH(X1, Beta)), c(n1, n2, n3))
Y <- array(rnorm(n1 * n2 * n3, Mu), c(n1, n2, n3))
system.time(fit <- glamlassoS(X, Y, V))

modelno <- length(fit$lambda)
plot(c(Beta), ylim = range(Beta, fit$coef[, modelno]), type = "h")
points(c(Beta))
lines(c(fit$coef[, modelno]), col = "red", type = "h")

###with non tensor design component Z
q <- 5
alpha <- matrix(rnorm(q)) * rbinom(q, 1, 0.5)
Z <- matrix(rnorm(n1 * n2 * n3 * q), n1 * n2 *n3, q) 
Y <- array(rnorm(n1 * n2 * n3, Mu + array(Z %*% alpha, c(n1, n2, n3))), c(n1, n2, n3))
system.time(fit <- glamlassoS(X, Y, V , Z))

modelno <- length(fit$lambda)
oldmfrow <- par()$mfrow
par(mfrow = c(1, 2))
plot(c(Beta), type="h", ylim = range(Beta, fit$coef[, modelno]))
points(c(Beta))
lines(fit$coef[ , modelno], col = "red", type = "h")
plot(c(alpha), type = "h", ylim = range(alpha, fit$alpha[, modelno]))
points(c(alpha))
lines(fit$alpha[ , modelno], col = "red", type = "h")
par(mfrow = oldmfrow)

################ poisson example
Beta <- matrix(rnorm(p1 * p2, 0, 0.1) * rbinom(p1 * p2, 1, 0.1), p1 , p2)
Mu <- V * array(RH(X2, RH(X1, Beta)), c(n1, n2, n3))
Y <- array(rpois(n1 * n2 * n3, exp(Mu)), dim = c(n1, n2, n3))
system.time(fit <- glamlassoS(X, Y, V, family = "poisson", nu = 0.1))

modelno <- length(fit$lambda)
plot(c(Beta), type = "h", ylim = range(Beta, fit$coef[, modelno]))
points(c(Beta))
lines(fit$coef[ , modelno], col = "red", type = "h")